Adaptive Control of Rikitake Two-Disk Dynamo System

International Journal of ChemTech Research CODEN (USA): IJCRGG ISSN: 0974-4290 Vol.8, No.8, pp 121-133, 2015 Adaptive Control of Rikitake Two-Disk Dynamo System Sundarapandian Vaidyanathan* R & D Centre, Vel Tech University, Avadi, Chennai, Tamil Nadu, India Abstract: Chaos theory has a manifold variety of applications in science and engineering. The frequent and irregular reversals of the earth s magnetic field has motivated a number of research studies involving electrical currents within the earth s molten core. One of the first such nonlinear models that exhibited the frequent and irregular reversals of the earth s magnetic field was Rikitake s two-disk dynamo system (1958). Rikitake two-disk dynamo system is a chaotic system that predated the pioneering work of Lorenz (1963). In this paper, we describe the dynamic equations and qualitative properties of the Rikitae two-disk dynamo chaotic system. We also derive new results for the adaptive control of the Rikitake two-disk dynamo chaotic system. MATLAB plots have been depicted to illustrate the phase portraits of the Rikitake two-disk dynamo chaotic attractor and the global chaos control of the Rikitake two-disk dynamo chaotic systems via adaptive control method. Keywords: Chaos, chaotic systems, chaos control, earth s magnetic field, electrical currents, Rikitake dynamo system, two-disk model, nonlinear model, adaptive control, stability. 1. Introduction A dynamical system is called chaotic if it satisfies the three properties: boundedness, infinite recurrence and sensitive dependence on initial conditions [1-2]. Chaos theory investigates the qualitative and numerical study of unstable aperiodic behaviour in deterministic nonlinear dynamical systems. In 1963, Lorenz [3] discovered a 3-D chaotic system when he was studying a 3-D weather model for atmospheric convection. After a decade, Rössler [4] discovered a 3-D chaotic system, which was constructed during the study of a chemical reaction. These classical chaotic systems paved the way to the discovery of many 3-D chaotic systems such as Arneodo system [5], Sprott systems [6], Chen system [7], Lü-Chen system [8], Cai system [9], Tigan system [10], etc. Many new chaotic systems have been also discovered in the recent years like Sundarapandian systems [11, 12], Vaidyanathan systems [13-43], Pehlivan system [44], Pham system [45], etc. Recently, there is significant result in the chaos literature in the synchronization of physical and chemical systems. A pair of systems called master and slave systems are considered for the synchronization process and the design goal is to device a feedback mechanism so that the trajectories of the slave system asymptotically track the trajectories of the master system. Because of the butterfly effect which causes exponential divergence of two trajectories of the system starting from nearby initial conditions, the synchronization of chaotic systems is seemingly a challenging research problem. In control theory, active control method is used when the parameters are available for measurement [46-65]. Adaptive control is a popular control technique used for stabilizing systems when the system parameters are unknown [66-79]. There are also other popular methods available for control and synchronization of systems such as backstepping control method [80-86], sliding mode control method [87-98], etc.

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 122 Recently, chaos theory is found to have important applications in several areas such as chemistry [99-107], biology [108-125], memristors [126-128], electrical circuits [129], etc. The frequent and irregular reversals of the earth s magnetic field has motivated a number of research studies involving electrical currents within the earth s molten core. One of the first such nonlinear models that exhibited the frequent and irregular reversals of the earth s magnetic field was Rikitake two-disk dynamo system [130]. Rikitake two-disk dynamo system (1958) is a classical chaotic system that predated the pioneering work of Lorenz (1963). First, this research paper details the dynamic equations of the Rikitake two-disk dynamo system [130] and discusses its qualitative properties. This paper also derives new results for the global chaos control of the Rikitake two-disk dynamo system via adaptive control method. MATLAB plots are shown to depict the phase portraits and global chaos control of the Rikitake two-disk dynamo system. 2. Rikitake Two-Disk Dynamo Chaotic System Rikitake two-disk dynamo chaotic system [130] is governed by the system model ìx& 1 =- ax1+ xx 2 3 íx& 2 =- ax2 + x1( x3 -b) (1) îx& 3= 1-xx 1 2 where x 1, x 2, x 3 are the states and a, bare constant positive parameters. The parameter a represents the resistive dissipation and the parameter b represents the difference in the angular velocities of the two disks. We note that the Rikitake two-disk dynamo chaotic system (1) has the same number of terms as the Lorenz chaotic system, but with one additional nonlinearity. The Rikitake two-disk dynamo system (1) is chaotic when the system parameters are chosen as a= 1, b= 1 (2) For numerical simulations, we take the initial conditions x (0) = 1.0, x (0) = 0, x (0) = 0.8 (3) 1 2 3 Figure 1 shows the 3-D phase portrait of the Rikitake two-disk dynamo system (1). Figures 2-4 show the 2-D projections of the Rikitake two-disk dynamo system on the ( x1, x2), ( x2, x3) and ( x1, x3) planes, respectively. Figure 1. The 3-D phase portrait of the Rikitake two-disk dynamo system

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 123 Figure 2. The 2-D projection of the Rikitake two-disk dynamo system on the ( x1, x2) plane Figure 3. The 2-D projection of the Rikitake two-disk dynamo system on the ( x2, x3) plane

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 124 Figure 4. The 2-D projection of the Rikitake two-disk dynamo system on the ( x1, x 3) plane The Lyapunov exponents of the Rikitake two-disk dynamo system (1) are numerically found as L === 0.12749, L 0, L - 2.12704 (4) 1 2 3 From the LE spectrum (4), it is immediate that the Rikitake two-disk dynamo system (1) is a chaotic system and the Maximal Lyapunov Exponent (MLE) of the Rikitake dynamo system (1) is L 1 = 0.12749. Since the sum of the Lyapunov exponents in (4) is negative, it follows that the Rikitake two-disk dynamo system (1) is dissipative. Also, the Lyapunov dimension of the Rikitake two-disk dynamo system (1) is derived as D L L + L = + = L 1 2 2 2.0599 3 3. Global Chaos Control of the Rikitake Two-Disk Dynamo Chaotic System via Adaptive Control Method In this section, we use adaptive control method to achieve global chaos control of the Rikitake two-disk dynamo chaotic system with unknown parameters. We use Lyapunov stability theory [131] to prove the main adaptive control result derived in this section. Thus, we consider the controlled Rikitake two-disk dynamo dynamics given by ìx& 1 =- ax1+ x2x3 + u1 íx& 2 =- ax2 + x1( x3- b) + u2 (6) îx& 3 = 1- x1x2 + u3 In ((6), x1, x2, x3are the states of the Rikitake two-disk dynamo system and abare, unknown parameters. The design of the control goal is to find nonlinear feedback controls u1, u2, u3using estimates of the unknown system parameters so as to regulate the states x1, x2, x3of the Rikitake two-disk dynamo chaotic system (6) to constant values or set-point controls given by a1, a2, a3, respectively. Thus, the output regulation errors are defined by ìe1() t = x1() t -a1 íe2() t = x2() t -a2 (7) îe3() t = x3() t -a3 (5)

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 125 The error dynamics is obtained as ìe& 1 =- a( e1+ a1) + ( e2 + a2)( e3 + a3) + u1 íe& 2 =- a( e2 + a2) + ( e1+ a1)( e3+ a3- b) + u2 (8) îe& 3 = 1 -( e1+ a1)( e2 + a2) + u3 We consider the adaptive controller defined by ìu1= at ˆ( )( e1+ a1) -( e2+ a2)( e3+ a3) -ke 1 1 íu ˆ 2= at ˆ( )( e2+ a2) -( e1+ a1)( e3+ a3-bt ( )) -ke 2 2 (9) u3=- 1 + ( e1+ a1)( e2+ a2) -ke î 3 3 where k 1, k 2, k 3 are positive gain constants. Substituting (9) into (8), we get the closed-loop error dynamics as ìe& 1=-[ a- at ˆ( )]( e1+ a1) -ke 1 1 íe& ˆ ˆ 2=-[ a- a( t)]( e2+ a2) -[ b- b( t)]( e1+ a1) -k2e2 (10) e& 3=-ke î 3 3 We define the parameter estimation errors as ìe () ˆ a t = a-at () í (11) e() ˆ î b t = b-bt () Using (11), the closed-loop system (10) can be simplified as ìe& 1= - ea( e1+ a1) -ke 1 1 íe& 2=- ea( e2+ a2) - eb( e1+ a1) -ke 2 2 (12) îe& 3=-ke 3 3 Differentiating (11) with respect to time, we get ì e& () ˆ a t =-at &() í (13) e() ˆ& î & b t =-bt () Next, we consider the candidate Lyapunov function defined by 1 2 2 2 2 2 V( e1, e2, e3, ea, eb) = ( e1 + e2 + e3 + ea + eb) (14) 2 Differentiating (14) along the trajectories of (12) and (13), we get the following dynamics 2 2 2 V& =-ke 1 1 -k2e2 - k3e3 + e é 1( 1 1) 2( 2 2) ˆ ˆ a - e e + a - e e + a - a& ù e é & + b - e2( e1+ a1) -bù (15) ë û êë úû In view of (15), we take the following parameter update law: ì a& ˆ() t =- e1( e1+ a1) - e2( e2 + a2) í (16) & îbˆ() t =- e2( e1+ a1) Next, we state and prove the main result of this section. Theorem 1. The adaptive control law (9) and the parameter update law (16) achieve global chaos control of the Rikitake two-disk dynamo system (6) by regulating the states x1(), t x2(), t x3() t so as to track the set-point controls a1, a2, a3, respectively, where k 1, k 2, k 3 are positive gain constants. Proof. The result is proved using Lyapunov stability theory [131]. The quadratic Lyapunov function V defined by (14) is positive definite on R 5. Substituting the parameter update law (16) into (15), we get the time derivative of V as

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 126 V & =-ke -k e -k e 2 2 2, 1 1 2 2 3 3 (17) which is negative semi-definite on R 5. Thus, by Barbalat s lemma in Lyapunov stability theory [131], it follows that the closed-loop error dynamics (12) is globally exponentially stable. Hence, we have shown that adaptive control law (9) and the parameter update law (16) achieve global chaos control of the Rikitake two-disk dynamo system (6) by regulating the states x1(), t x2(), t x3() t so as to track the set-point controls a1, a2, a3, respectively. This completes the proof. 4. Numerical Simulations -8 We use the classical fourth-order Runge-Kutta method with step-size h = 10 to solve the system of differential equations (6) and (7), when the adaptive control law (10) is implemented. We take the parameter values of the Rikitake two-disk dynamo chaotic systems as in the chaotic case, viz. a= 1, b= 1 (18) We take the gain constants as k1 = 6, k2 = 6, k3 = 6 (19) We take the initial values of the Rikitake dynamo system (6) as x1(0) = 7.5, x2(0) = 17.2, x3(0) = 23.4 (20) We take the set-point controls as a1 = 1, a2 = 2, a3 = 3 (21) We take the initial values of the parameter estimates as aˆ(0) = 3.4, bˆ (0) = 7.6 (22) Figures 5-7 show the output regulation of the Rikitake dynamo chaotic system (6). Figure 8 shows the time-history of the regulation errors e1, e2, e3. Figure 5. Output regulation of the state x () t 1

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 127 Figure 6. Output regulation of the state x () t 2 Figure 7. Output regulation of the state x () t 3 Figure 8. Time-history of the regulation errors e1(), t e2(), t e3() t

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 128 5. Conclusions In this paper, we described the dynamic equations and qualitative properties of the Rikitae two-disk dynamo chaotic system. We also derived new results for the global chaos control of the Rikitake two-disk dynamo chaotic system. MATLAB plots were depicted to illustrate the phase portraits of the Rikitake two-disk dynamo chaotic attractor and the output regulation of the Rikitake two-disk dynamo chaotic system via adaptive control method. References 1. Azar, A. T., and Vaidyanathan, S., Chaos Modeling and Control Systems Design, Studies in Computational Intelligence, Vol. 581, Springer, New York, USA, 2015. 2. Azar, A. T., and Vaidyanathan, S., Computational Intelligence Applications in Modeling and Control, Studies in Computational Intelligence, Vol. 575, Springer, New York, USA, 2015. 3. Lorenz, E. N., Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 1963, 20, 130-141. 4. Rössler, O. E., An equation for continuous chaos, Physics Letters A, 1976, 57, 397-398. 5. Arneodo, A., Coullet, P., and Tresser, C., Possible new strange attractors with spiral structure, Communications in Mathematical Physics, 1981, 79, 573-579. 6. Sprott, J. C., Some simple chaotic flows, Physical Review E, 1994, 50, 647-650. 7. Chen, G., and Ueta, T., Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 1999, 9, 1465-1466. 8. Lü, J., and Chen, G., A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 2002, 12, 659-661. 9. Cai, G., and Tan, Z., Chaos synchronization of a new chaotic system via nonlinear control, Journal of Uncertain Systems, 2007, 1, 235-240. 10. Tigan, G., and Opris, D., Analysis of a 3D chaotic system, Chaos, Solitons and Fractals, 2008, 36, 1315-1319. 11. Sundarapandian, V., and Pehlivan, I., Analysis, control, synchronization and circuit design of a novel chaotic system, Mathematical and Computer Modelling, 2012, 55, 1904-1915. 12. Sundarapandian, V., Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers, Journal of Engineering Science and Technology Review, 2013, 6, 45-52. 13. Vaidyanathan, S., A new six-term 3-D chaotic system with an exponential nonlinearity, Far East Journal of Mathematical Sciences, 2013, 79, 135-143. 14. Vaidyanathan, S., Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters, Journal of Engineering Science and Technology Review, 2013, 6, 53-65. 15. Vaidyanathan, S., A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities, Far East Journal of Mathematical Sciences, 2014, 84, 219-226. 16. Vaidyanathan, S., Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities, International Journal of Modelling, Identification and Control, 2014, 22, 41-53. 17. Vaidyanathan, S., and Madhavan, K., Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system, International Journal of Control Theory and Applications, 2013, 6, 121-137. 18. Vaidyanathan, S., Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities, European Physical Journal: Special Topics, 2014, 223, 1519-1529. 19. Vaidyanathan, S., Volos, C., Pham, V. T., Madhavan, K., and Idowu, B. A., Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities, Archives of Control Sciences, 2014, 24, 257-285. 20. Vaidyanathan, S., Generalised projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control, International Journal of Modelling, Identification and Control, 2014, 22, 207-217. 21. Vaidyanathan, S., and Azar, A.T., Analysis and control of a 4-D novel hyperchaotic system, Studies in Computational Intelligence, 2015, 581, 3-17. 22. Vaidyanathan, S., Volos, C., Pham, V.T., and Madhavan, K., Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation, Archives of Control Sciences, 2015, 25, 135-158.

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 129 23. Vaidyanathan, S., Volos, C., and Pham, V.T., Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation, Archives of Control Sciences, 2014, 24, 409-446. 24. Vaidyanathan, S., A ten-term novel 4-D hyperchaotic system with three quadratic nonlinearities and its control, International Journal of Control Theory and Applications, 2013, 6, 97-109. 25. Vaidyanathan, S., Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity, International Journal of Modelling, Identification and Control, 2015, 23, 164-172. 26. Vaidyanathan, S., Azar, A.T., Rajagopal, K., and Alexander, P., Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronisation via active control, International Journal of Modelling, Identification and Control, 2015, 23, 267-277. 27. Vaidyanathan, S., Qualitative analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with a quartic nonlinearity, International Journal of Control Theory and Applications, 2014, 7, 1-20. 28. Vaidyanathan, S., Qualitative analysis and control of an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities, International Journal of Control Theory and Applications, 2014, 7, 35-47. 29. Vaidyanathan, S., and Pakiriswamy, S., A 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive control, Journal of Engineering Science and Technology Review, 2015, 8, 52-60. 30. Vaidyanathan, S., Volos, C.K., and Pham, V.T., Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation, Journal of Engineering Science and Technology Review, 2015, 8, 181-191. 31. Vaidyanathan, S., Rajagopal, K., Volos, C.K., Kyprianidis, I.M., and Stouboulos, I.N., Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW, Journal of Engineering Science and Technology Review, 2015, 8, 130-141. 32. Pham, V.T., Volos, C.K., Vaidyanathan, S., Le, T.P., and Vu, V.Y., A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating, Journal of Engineering Science and Technology Review, 2015, 8, 205-214. 33. Pham, V.T., Volos, C.K., and Vaidyanathan, S., Multi-scroll chaotic oscillator based on a first-order delay differential equation, Studies in Computational Intelligence, 2015, 581, 59-72. 34. Vaidyanathan, S., Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N., and Pham, V.T., Analysis, adaptive control and anti-synchronization of a six-term novel jerk chaotic system with two exponential nonlinearities and its circuit simulation, Journal of Engineering Science and Technology Review, 2015, 8, 24-36. 35. Vaidyanathan, S., Volos, C.K., and Pham, V.T., Analysis, control, synchronization and SPICE implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium, Journal of Engineering Science and Technology Review, 2015, 8, 232-244. 36. Sampath, S., Vaidyanathan, S., Volos, C.K., and Pham, V.T., An eight-term novel four-scroll chaotic system with cubic nonlinearity and its circuit simulation, Journal of Engineering Science and Technology Review, 2015, 8, 1-6. 37. Vaidyanathan, S., A 3-D novel highly chaotic system with four quadratic nonlinearities, its adaptive control and anti-synchronization with unknown parameters, Journal of Engineering Science and Technology Review, 2015, 8, 106-115. 38. Vaidyanathan, S., Pham, V.-T., and Volos, C. K., A 5-D hyperchaotic Rikitake dynamo system with hidden attractors, European Physical Journal: Special Topics, 2015, 224, 1575-1592. 39. Pham, V.-T., Vaidyanathan, S., Volos, C. K., and Jafari, S., Hidden attractors in a chaotic system with an exponential nonlinear term, European Physical Journal: Special Topics, 2015, 224, 1507-1517. 40. Vaidyanathan, S., Hyperchaos, qualitative analysis, control and synchronisation of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities, International Journal of Modelling, Identification and Control, 2015, 23, 380-392. 41. Vaidyanathan, S., and Azar, A. T., Analysis, control and synchronization of a nine-term 3-D novel chaotic system, Studies in Computational Intelligence, 2015, 581, 19-38. 42. Vaidyanathan, S., and Volos, C., Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system, Archives of Control Sciences, 2015, 25, 279-299. 43. Vaidyanathan, S., Analysis, control and synchronization of a 3-D novel jerk chaotic system with two

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 130 quadratic nonlinearities, Kyungpook Mathematical Journal, 2015, 55, 563-586. 44. Pehlivan, I., Moroz, I. M., and Vaidyanathan, S., Analysis, synchronization and circuit design of a novel butterfly attractor, Journal of Sound and Vibration, 2014, 333, 5077-5096. 45. Pham, V. T., Volos, C., Jafari, S., Wang, X., and Vaidyanathan, S., Hidden hyperchaotic attractor in a novel simple memristic neural network, Optoelectronics and Advanced Materials Rapid Communications, 2014, 8, 1157-1163. 46. Sundarapandian, V., Output regulation of Van der Pol oscillator, Journal of the Institution of Engineers (India): Electrical Engineering Division, 88, 20-24, 2007. 47. Sundarapandian, V., Output regulation of the Lorenz attractor, Far East Journal of Mathematical Sciences, 2010, 42, 289-299. 48. Vaidyanathan, S., and Rajagopal, K., Anti-synchronization of Li and T chaotic systems by active nonlinear control, Communications in Computer and Information Science, 2011, 198, 175-184. 49. Vaidyanathan, S., and Rasappan, S., Global chaos synchronization of hyperchaotic Bao and Xu systems by active nonlinear control, Communications in Computer and Information Science, 2011, 198, 10-17. 50. Vaidyanathan, S., Output regulation of the unified chaotic system, Communications in Computer and Information Science, 2011, 198, 1-9. 51. Vaidyanathan, S., and Rajagopal, K., Global chaos synchronization of hyperchaotic Pang and Wang systems by active nonlinear control, 2011, 198, 84-93. 52. Vaidyanathan, S., Hybrid chaos synchronization of Liu and Lu systems by active nonlinear control, Communications in Computer and Information Science, 2011, 204, 1-10. 53. Sarasu, P., and Sundarapandian, V., Active controller design for generalized projective synchronization of four-scroll chaotic systems, International Journal of Systems Signal Control and Engineering Application, 2011, 4, 26-33. 54. Vaidyanathan, S., and Rasappan, S., Hybrid synchronization of hyperchaotic Qi and Lu systems by nonlinear control, Communications in Computer and Information Science, 2011, 131, 585-593. 55. Vaidyanathan, S., and Rajagopal, K., Hybrid synchronization of hyperchaotic Wang-Chen and hyperchaotic Lorenz systems by active non-linear control, International Journal of Systems Signal Control and Engineering Application, 2011, 4, 55-61. 56. Vaidyanathan, S., Output regulation of Arneodo-Coullet chaotic system, Communications in Computer and Information Science, 2011, 133, 98-107. 57. Sarasu, P., and Sundarapandian, V., The generalized projective synchronization of hyperchaotic Lorenz and hyperchaotic Qi systems via active control, International Journal of Soft Computing, 2011, 6, 216-223. 58. Vaidyanathan, S., and Pakiriswamy, S., The design of active feedback controllers for the generalized projective synchronization of hyperchaotic Qi and hyperchaotic Lorenz systems, Communications in Computer and Information Science, 2011, 245, 231-238. 59. Sundarapandian, V., and Karthikeyan, R., Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control, Journal of Engineering and Applied Sciences, 2012, 7, 254-264. 60. Vaidyanathan, S., and Pakiriswamy, S., Generalized projective synchronization of double-scroll chaotic systems using active feedback control, Lecture Notes of the Institute for Computer Sciences, Social- Informatics and Telecommunications Engineering, 2012, 84, 111-118. 61. Pakiriswamy, S., and Vaidyanathan, S., Generalized projective synchronization of three-scroll chaotic systems via active control, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, 2012, 85, 146-155. 62. Karthikeyan, R., and Sundarapandian, V., Hybrid chaos synchronization of four-scroll systems via active control, Journal of Electrical Engineering, 2014, 65, 97-103. 63. Vaidyanathan, S., Azar, A. T., Rajagopal, K., and Alexander, P., Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronisation via active control, International Journal of Modelling, Identification and Control, 2015, 23, 267-277. 64. Yassen, M. T., Chaos synchronization between two different chaotic systems using active control, Chaos, Solitons and Fractals, 2005, 23, 131-140. 65. Jia, N., and Wang, T., Chaos control and hybrid projective synchronization for a class of new chaotic systems, Computers and Mathematics with Applications, 2011, 62, 4783-4795. 66. Vaidyanathan, S., and Rajagopal, K., Global chaos synchronization of Lu and Pan systems by adaptive nonlinear control, Communication in Computer and Information Science, 2011, 205, 193-202.

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 131 67. Vaidyanathan, S., Adaptive controller and synchronizer design for the Qi-Chen chaotic system, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunication Engineering, 2012, 85, 124-133. 68. Sundarapandian, V., Adaptive control and synchronization design for the Lu-Xiao chaotic system, Lectures on Electrical Engineering, 2013, 131, 319-327. 69. Vaidyanathan, S., Analysis, control and synchronization of hyperchaotic Zhou system via adaptive control, Advances in Intelligent Systems and Computing, 2013, 177, 1-10. 70. Vaidyanathan, S., and Rajagopal, K., Global chaos synchronization of Lü and Pan systems by adaptive nonlinear control, Communications in Computer and Information Science, 2011, 205, 193-202. 71. Sundarapandian, V., and Karthikeyan, R., Anti-synchronization of Lü and Pan systems by adaptive nonlinear control, European Journal of Scientific Research, 2011, 64, 94-106. 72. Sundarapandian, V., and Karthikeyan, R., Anti-synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive control, International Journal of Systems Signal Control and Engineering Application, 2011, 4, 18-25. 73. Sundarapandian, V., and Karthikeyan, R., Adaptive anti-synchronization of uncertain Tigan and Li systems, Journal of Engineering and Applied Sciences, 2012, 7, 45-52. 74. Sarasu, P., and Sundarapandian, V., Generalized projective synchronization of three-scroll chaotic systems via adaptive control, European Journal of Scientific Research, 2012, 72, 504-522. 75. Vaidyanathan, S., and Rajagopal, K., Global chaos synchronization of hyperchaotic Pang and hyperchaotic Wang systems via adaptive control, International Journal of Soft Computing, 2012, 7, 28-37. 76. Sarasu, P., and Sundarapandian, V., Generalized projective synchronization of two-scroll systems via adaptive control, International Journal of Soft Computing, 2012, 7, 146-156. 77. Sarasu, P., and Sundarapandian, V., Adaptive controller design for the generalized projective synchronization of 4-scroll systems, International Journal of Systems Signal Control and Engineering Application, 2012, 5, 21-30. 78. Vaidyanathan, S., Anti-synchronization of Sprott-L and Sprott-M chaotic systems via adaptive control, International Journal of Control Theory and Applications, 2012, 5, 41-59. 79. Vaidyanathan, S., and Pakiriswamy, S., Generalized projective synchronization of six-term Sundarapandian chaotic systems by adaptive control, International Journal of Control Theory and Applications, 2013, 6, 153-163. 80. Rasappan, S., and Vaidyanathan, S., Hybrid synchronization of n-scroll chaotic Chua circuits using adaptive backstepping control design with recursive feedback, Malaysian Journal of Mathematical Sciences, 2013, 7, 219-246. 81. Suresh, R., and Sundarapandian, V., Global chaos synchronization of a family of n-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback, Far East Journal of Mathematical Sciences, 2013, 73, 73-95. 82. Rasappan, S., and Vaidyanathan, S., Hybrid synchronization of n-scroll Chua and Lur e chaotic systems via backstepping control with novel feedback, Archives of Control Sciences, 2012, 22, 343-365. 83. Rasappan, S., and Vaidyanathan, S., Global chaos synchronization of WINDMI and Coullet chaotic systems using adaptive backstepping control design, Kyungpook Mathematical Journal, 2014, 54, 293-320. 84. Vaidyanathan, S., and Rasappan, S., Global chaos synchronization of n-scroll Chua circuit and Lur e system using backstepping control design with recursive feedback, Arabian Journal for Science and Engineering, 2014, 39, 3351-3364. 85. Vaidyanathan, S., Idowu, B. A., and Azar, A. T., Backstepping controller design for the global chaos synchronization of Sprott s jerk systems, Studies in Computational Intelligence, 2015, 581, 39-58. 86. Vaidyanathan, S., Volos, C. K., Rajagopal, K., Kyprianidis, I. M., and Stouboulos, I. N., Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation, Journal of Engineering Science and Technology Review, 2015, 8, 74-82. 87. Vaidyanathan, S., and Sampath, S., Global chaos synchronization of hyperchaotic Lorenz systems by sliding mode control, Communications in Computer and Information Science, 2011, 205, 156-164. 88. Sundarapandian, V., and Sivaperumal, S., Sliding controller design of hybrid synchronization of fourwing chaotic systems, International Journal of Soft Computing, 2011, 6, 224-231. 89. Vaidyanathan, S., and Sampath, S., Anti-synchronization of four-wing chaotic systems via sliding mode

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 132 control, International Journal of Automation and Computing, 2012, 9, 274-279. 90. Vaidyanathan, S., and Sampath, S., Sliding mode controller design for the global chaos synchronization of Coullet systems, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, 2012, 84, 103-110. 91. Vaidyanathan, S., and Sampath, S., Hybrid synchronization of hyperchaotic Chen systems via sliding mode control, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, 2012, 85, 257-266. 92. Vaidyanathan, S., Global chaos control of hyperchaotic Liu system via sliding control method, International Journal of Control Theory and Applications, 2012, 5, 117-123. 93. Vaidyanathan, S., Sliding mode control based global chaos control of Liu-Liu-Liu-Su chaotic system, International Journal of Control Theory and Applications, 2012, 5, 15-20. 94. Vaidyanathan, S., Global chaos synchronisation of identical Li-Wu chaotic systems via sliding mode control, International Journal of Modelling, Identification and Control, 2014, 22, 170-177. 95. Vaidyanathan, S., and Azar, A. T., Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan-Madhavan chaotic systems, Studies in Computational Intelligence, 2015, 576, 527-547. 96. Vaidyanathan, S., and Azar, A. T., Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan chaotic systems, Studies in Computational Intelligence, 2015, 576, 549-569. 97. Vaidyanathan, S., Sampath, S., and Azar, A. T., Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system, International Journal of Modelling, Identification and Control, 2015, 23, 92-100. 98. Li, H., Liao, X., Li, C., and Li, C., Chaos control and synchronization via a novel chatter free sliding mode control strategy, Neurocomputing, 2011, 74, 3212-3222. 99. Vaidyanathan, S., Adaptive synchronization of chemical chaotic reactors, International Journal of ChemTech Research, 2015, 8, 612-621. 100. Vaidyanathan, S., Adaptive control of a chemical chaotic reactor, International Journal of PharmTech Research, 2015, 8, 377-382. 101. Vaidyanathan, S., Dynamics and control of Brusselator chemical reaction, International Journal of ChemTech Research, 2015, 8, 740-749. 102. Vaidyanathan, S., Anti-synchronization of Brusselator chemical reaction systems via adaptive control, International Journal of ChemTech Research, 2015, 8, 759-768. 103. Vaidyanathan, S., Dynamics and control of Tokamak system with symmetric and magnetically confined plasma, International Journal of ChemTech Research, 2015, 8, 795-803. 104. Vaidyanathan, S., Synchronization of Tokamak systems with symmetric and magnetically confined plasma via adaptive control, International Journal of ChemTech Research, 2015, 8, 818-827. 105. Vaidyanathan, S., A novel chemical chaotic reactor system and its adaptive control, International Journal of ChemTech Research, 2015, 8, 146-158. 106. Vaidyanathan, S., Adaptive synchronization of novel 3-D chemical chaotic reactor systems, International Journal of ChemTech Research, 2015, 8, 159-171. 107. Vaidyanathan, S., Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method, International Journal of ChemTech Research, 2015, 8, 209-221. 108. Garfinkel, A., Spano, M.L., Ditto, W.L., and Weiss, J.N., Controlling cardiac chaos, Science, 1992, 257, 1230-1235. 109. May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 259-267. 110. Vaidyanathan, S., Adaptive backstepping control of enzymes-substrates system with ferroelectric behaviour in brain-waves, International Journal of PharmTech Research, 2015, 8, 256-261. 111. Vaidyanathan, S., Adaptive biological control of generalized Lotka-Volterra three species biological system, International Journal of PharmTech Research, 2015, 8, 622-631. 112. Vaidyanathan, S., 3-cells cellular neural network (CNN) attractor and its adaptive biological control, International Journal of PharmTech Research, 2015, 8, 632-640. 113. Vaidyanathan, S., Adaptive synchronization of generalized Lotka-Volterra three species biological systems, International Journal of PharmTech Research, 2015, 8, 928-937. 114. Vaidyanathan, S., Synchronization of 3-cells cellular neural network (CNN) attractors via adaptive control method, International Journal of PharmTech Research, 2015, 8, 946-955. 115. Vaidyanathan, S., Chaos in neurons and adaptive control of Birkhoff-Shaw strange chaotic attractor,

Sundarapandian Vaidyanathan /Int.J. ChemTech Res. 2015,8(8),pp 121-133. 133 International Journal of PharmTech Research, 2015, 8, 956-963. 116. Vaidyanathan, S., Adaptive chaotic synchronization of enzymes-substrates system with ferroelectric behaviour in brain waves, International Journal of PharmTech Research, 2015, 8, 964-973. 117. Vaidyanathan, S., Lotka-Volterra population biology models with negative feedback and their ecological monitoring, International Journal of PharmTech Research, 2015, 8, 974-981. 118. Vaidyanathan, S., Chaos in neurons and synchronization of Birkhoff-Shaw strange chaotic attractors via adaptive control, International Journal of PharmTech Research, 2015, 8, 1-11. 119. Vaidyanathan, S., Lotka-Volterra two species competitive biology models and their ecological monitoring, International Journal of PharmTech Research, 2015, 8, 32-44. 120. Vaidyanathan, S., Coleman-Gomatam logarithmic competitive biology models and their ecological monitoring, International Journal of PharmTech Research, 2015, 8, 94-105. 121. Vaidyanathan, S., Output regulation of the forced Van der Pol chaotic oscillator via adaptive control method, International Journal of PharmTech Research, 2015, 8, 106-116. 122. Vaidyanathan, S., Adaptive control of the FitzHugh-Nagumo chaotic neuron model, International Journal of PharmTech Research, 2015, 8, 117-127. 123. Vaidyanathan, S., Global chaos synchronization of the forced Van der Pol chaotic oscillators via adaptive control method, International Journal of PharmTech Research, 2015, 8, 156-166. 124. Vaidyanathan, S., Adaptive synchronization of the identical FitzHugh-Nagumo chaotic neuron models, International Journal of PharmTech Research, 2015, 8, 167-177. 125. Vaidyanathan, S., Global chaos synchronization of the Lotka-Volterra biological systems with four competitive species via active control, International Journal of PharmTech Research, 2015, 8, 206-217. 126. Pham, V.-T., Volos, C. K., Vaidyanathan, S., and Vu, V. Y., A memristor-based hyperchaotic system with hidden attractors: dynamics, synchronization and circuital emulating, Journal of Engineering Science and Technology Review, 2015, 8, 205-214. 127. Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Tlelo-Cuautle, E., and Vaidyanathan, S., Memristor: A new concept in synchronization of coupled neuromorphic circuits, Journal of Engineering Science and Technology Review, 2015, 8, 157-173. 128. Pham, V.-T., Volos, C., Jafari, S., Wang, X., and Vaidyanathan, S., Hidden hyperchaotic attractor in a novel simple memristive neural network, Optoelectronics and Advanced Materials, Rapid Communications, 2014, 8, 1157-1163. 129. Volos, C. K., Pham, V.-T., Vaidyanathan, S., Kyprianidis, I. M., and Stouboulos, I. N., Synchronization phenomena in coupled Colpitts circuits, Journal of Engineering Science and Technology Review, 2015, 8, 142-151. 130. Rikitake, T., Oscillations of a system of disk dynamos, Mathematical Proceedings of the Cambridge Philosophical Society, 1958, 54, 89-105. 131. Khalil, H. K., Nonlinear Systems, Third Edition, Prentice Hall, New Jersey, USA, 2002. *****